Home
Class 12
MATHS
Two variable chords A Ba n dB C of a cir...

Two variable chords `A Ba n dB C` of a circle `x^2+y^2=r^2` are such that `A B=B C=r` . Find the locus of the point of intersection of tangents at `Aa n dCdot`

Text Solution

Verified by Experts

The correct Answer is:
`x^(2)+y^(2)=4r^(2)`
Let the point of intersection of tangents at A and be `P -= (x,y)` .
Since `AB =AO =BO =r`, we get
`/_ AOB =60^(@)`

or `(PA)/(r) tan 60^(@) = sqrt (3)`
or `sqrt(x^(2)+y^(2)-r^(2))=sqrt(3)r`
Therefore, `x^(2)+y^(2)=4r^(2)` is the locus of the point P.
Promotional Banner

Topper's Solved these Questions

  • CIRCLE

    CENGAGE PUBLICATION|Exercise CONCEPT APPLICATION EXERCISE 4.16|3 Videos

  • CIRCLE

    CENGAGE PUBLICATION|Exercise CONCEPT APPLICATION EXERCISE 4.17|2 Videos

  • CIRCLE

    CENGAGE PUBLICATION|Exercise CONCEPT APPLICATION EXERCISE 4.20|1 Videos

  • BINOMIAL THEOREM

    CENGAGE PUBLICATION|Exercise Comprehension|11 Videos

  • CIRCLES

    CENGAGE PUBLICATION|Exercise Comprehension Type|8 Videos

Similar Questions

Explore conceptually related problems

From an arbitrary point P on the circle x^2+y^2=9 , tangents are drawn to the circle x^2+y^2=1 , which meet x^2+y^2=9 at A and B . The locus of the point of intersection of tangents at A and B to the circle x^2+y^2=9 is (a) x^2+y^2=((27)/7)^2 (b) x^2-y^2((27)/7)^2 (c) y^2-x^2=((27)/7)^2 (d) none of these

If a tangent to the parabola y^2 = 4ax intersects the x^2/a^2+y^2/b^2= 1 at A and B , then the locus of the point of intersection of tangents at A and B to the ellipse is

The equation of the locus of the middle point of a chord of the circle x^2+y^2=2(x+y) such that the pair of lines joining the origin to the point of intersection of the chord and the circle are equally inclined to the x-axis is (a) x+y=2 (b) x-y=2 (c) 2x-y=1 (d) none of these

Find the locus of the point of intersection of the perpendicular tangents of the curve y^2+4y-6x-2=0 .

A variable chord is drawn through the origin to the circle x^2+y^2-2a x=0 . Find the locus of the center of the circle drawn on this chord as diameter.

A circle with center at the origin and radius equal to a meets the axis of x at Aa n dB. P(alpha) and Q(beta) are two points on the circle so that alpha-beta=2gamma , where gamma is a constant. Find the locus of the point of intersection of A P and B Qdot

In a triangle A B C ,Da n dE are points on B Ca n dA C , respectivley, such that B D=2D Ca n dA E=3E Cdot Let P be the point of intersection of A Da n dB Edot Find B P//P E using the vector method.

From the variable point A on circle x^2+y^2=2a^2, two tangents are drawn to the circle x^2+y^2=a^2 which meet the curve at B and Cdot Find the locus of the circumcenter of A B Cdot

A variable plane passes through a fixed point (alpha,beta,gamma) and meets the axes at A ,B ,a n dCdot show that the locus of the point of intersection of the planes through A ,Ba n dC parallel to the coordinate planes is alphax^(-1)+betay^(-1)+gammaz^(-1)=1.

If the tangents are drawn to the circle x^2+y^2=12 at the point where it meets the circle x^2+y^2-5x+3y-2=0, then find the point of intersection of these tangents.